Subfield codes were first considered in [1]} and [2]} without using the name “subfield codes". The definition of subfield codes was first given by [3]} and a Magma function for subfield codes is actual operated. Recently, Ding and Heng in [4]} proved some basic results about subfield codes of linear codes and gave their trace representation.
Assume that \(\mathbb {F}_{p^m}=\lbrace x_1,x_2,\cdots ,x_{p^m}\rbrace \)  and \(f(x)\)  is a polynomial over \(\mathbb {F}_{p^m}\) . Let \(\mathcal {C}\)  be a \([p^m+1,3]\)  code with the generator matrix
\(\begin{split}G = \left(\begin{array}{cccccc}f(x_1) & f(x_2) & \cdots & f(x_{p^m}) & 1 \\x_1 & x_2 & \cdots & x_{p^m} & 0 \\1 & 1 & \cdots & 1 & 0\end{array}\right).\end{split}\) 
